Final answer:
To find the critical point and the interval on which the given function is increasing or decreasing, we first need to find the derivative of the function. The derivative of f(x) = 6ln(4x) - 2x is f'(x) = 6/x - 2. The critical point is x = 3, and the function is increasing on the interval (0, 3) and decreasing on the interval (3, ∞). The critical point is a local maximum.
Step-by-step explanation:
To find the critical point and the interval on which the given function is increasing or decreasing, we first need to find the derivative of the function. The derivative of f(x) = 6ln(4x) - 2x is f'(x) = 6/x - 2. To find the critical point, we set f'(x) equal to zero and solve for x. 6/x - 2 = 0, 6/x = 2, x = 3. Therefore, the critical point is x = 3.
To determine the intervals of increasing and decreasing, we can use the first derivative test. The first derivative is positive when x is less than 3, and negative when x is greater than 3. This means the function is increasing on the interval (0, 3) and decreasing on the interval (3, ∞).
Applying the first derivative test, we evaluate the derivative at the critical point (x = 3). Since the derivative changes sign from positive to negative, the critical point is a local maximum.